What kind of overshoes can I use with a large touring SPD cycling shoe such as the Giro Rumble VR?

You need overshoes specifically designed for MTB shoes. Note that most of them would say Road Compatible: Yes, MTB Compatible: Yes but they are typically optimised for road shoes. I have the Shimano S3000X NPU + overshoes which are marketed as MTB overshoes and they are great, although it looks like they are hard to find nowadays in reasonable sizes.

I have the same shoes in UK12/EU47/US13. I have some XL Madison Sportive overshoes, and they've survived a couple of years. I recently ripped them putting them on over my other shoes (an old pair of sadly discontinued Specialized Cadet), but not beyond repair, and it might have been caused by the bits that touch the ground being rather worn and therefore not reinforcing the upper.

It is a process of jiggling and pulling to get them over the shoes - I allow an extra 5 minutes to get ready if I know I'm putting them on (it doesn't take quite that long)

I also, in my commuting bag, have a pair of Eager brand overshoes, that owe more to hiking gaiters. They're floppy rather than stretchy, and don't really look the part, but they're quick to put on and off, and work fairly well. I've only been lightly mocked for them once, when at a gathering of proper cyclists, and by someone who was probably waiting for the chance.

Here are the 2 shoe covers: floppy Eager on the left, tight Madison on the right.